A quantum system $\mathcal{Q}$ is characterized by a single potential $v$ and its eigenstates. While $v$ is usually postulated for a given physical problem, it represents the interaction with an implicit environment $\mathcal{E}$ . We use the exact factorization to show how $v$ emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential ${v}_{j}$ and state ${\ensuremath{\chi}}_{j}$ of $\mathcal{Q}$ . Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of ${v}_{j}$ , even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting $\mathcal{Q}$ is much smaller than for exciting $\mathcal{E}$ , the potentials ${v}_{j}$ are similar in shape and differ only by sharp spikes. An excitation of $\mathcal{S}$ can then be viewed as an excitation of $\mathcal{Q}$ with its environment being unaffected, and $\mathcal{Q}$ is approximately described by a single spikeless potential $v$ and its eigenstates. Second, ${\ensuremath{\chi}}_{j}$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\ensuremath{\chi}}_{j}$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.A quantum system $\mathcal{Q}$ is characterized by a single potential $v$ and its eigenstates. While $v$ is usually postulated for a given physical problem, it represents the interaction with an implicit environment $\mathcal{E}$. We use the exact factorization to show how $v$ emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential ${v}_{j}$ and state ${\ensuremath{\chi}}_{j}$ of $\mathcal{Q}$. Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of ${v}_{j}$, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting $\mathcal{Q}$ is much smaller than for exciting $\mathcal{E}$, the potentials ${v}_{j}$ are similar in shape and differ only by sharp spikes. An excitation of $\mathcal{S}$ can then be viewed as an excitation of $\mathcal{Q}$ with its environment being unaffected, and $\mathcal{Q}$ is approximately described by a single spikeless potential $v$ and its eigenstates. Second, ${\ensuremath{\chi}}_{j}$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\ensuremath{\chi}}_{j}$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.

Physical Review ResearchAug 27, 20214 citations

Quantum solutions for the harmonic-parabola potential system

Kyu Hwang Yeon, Shou Zhang, Yong Dae Kim +2 more

The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.

Physical Review AMar 16, 20008 citations

States in Quantum Mechanics

L. E. Ballentine

Compendium of Quantum PhysicsJan 1, 20090 citations

Phase Regularization and Additional Potential in Quantum Systems at the Potential Barrier to Maintain Adiabatic Condition

Anggen Dari, Iwan Setiawan, Andik Purwanto

This research is theoretical research with a literature study that examines methods to maintain characteristics of electrons when moving in a quantum system at the potential barrier. Attempts to maintain the characteristics of electrons in quantum systems are known as adiabatic. The method used in this research is the fast-forward method. This method was first introduced by Masuda and Nakamura in 2010. The fast-forward method is applied to barrier potential in case of electron energy larger and smaller than the potential barrier. This research focuses on preserving the characteristics of electrons by determining the regularization phase and additional potential of the quantum system at the barrier potential. The wave function solution of the system at the potential barrier is approximated by the Schrödinger equation into three regions for each case. The wave function of each region is regularized to be an adiabatic wave function and an additional term in the form of regularization phase (θ) is obtained. Given the regularized Hamiltonian, the additional potential (Ṽ) is obtained. The obtained regularization phase and additional potential ensure the quantum system at the potential barrier is in the same state from the initial state to the final stateThis research is theoretical research with a literature study that examines methods to maintain characteristics of electrons when moving in a quantum system at the potential barrier. Attempts to maintain the characteristics of electrons in quantum systems are known as adiabatic. The method used in this research is the fast-forward method. This method was first introduced by Masuda and Nakamura in 2010. The fast-forward method is applied to barrier potential in case of electron energy larger and smaller than the potential barrier. This research focuses on preserving the characteristics of electrons by determining the regularization phase and additional potential of the quantum system at the barrier potential. The wave function solution of the system at the potential barrier is approximated by the Schrödinger equation into three regions for each case. The wave function of each region is regularized to be an adiabatic wave function and an additional term in the form of regularization phase (θ) is obtained. Given the regularized Hamiltonian, the additional potential (Ṽ) is obtained. The obtained regularization phase and additional potential ensure the quantum system at the potential barrier is in the same state from the initial state to the final state

Jurnal Penelitian Pendidikan IPAFeb 28, 20240 citations

The Structure of Quantum States

Gary E. Bowman

Oxford University Press eBooksNov 1, 20070 citations

A quantum system

\mathcal{Q}

is characterized by a single potential

v

and its eigenstates. While

v

is usually postulated for a given physical problem, it represents the interaction with an implicit environment

\mathcal{E}

. We use the exact factorization to show how

v

emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential

{v}_{j}

and state ${\ensuremath{\chi}}_{j}$ of

\mathcal{Q}

. Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of

{v}_{j}

, even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting

\mathcal{Q}

is much smaller than for exciting

\mathcal{E}

, the potentials

{v}_{j}

are similar in shape and differ only by sharp spikes. An excitation of

\mathcal{S}

can then be viewed as an excitation of

\mathcal{Q}

with its environment being unaffected, and

\mathcal{Q}

is approximately described by a single spikeless potential

v

and its eigenstates. Second, ${\ensuremath{\chi}}_{j}$ can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of ${\ensuremath{\chi}}_{j}$ being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.